| | Ivars Peterson's MathTrek |
September 7, 1998
In the August SIAM Journal of Applied Mathematics, Sergio Rinaldi of the Politecnico de Milano in Italy brings differential equations to bear on an area where such techniques have been rarely applied -- the dynamics of love. He proposes a model that he claims captures the emotional ups-and-downs of the unrequited love of the celebrated poet and scholar Francesco Petrarca (1304-74), also known as Petrarch, for Laura, a beautiful but married lady of Avignon, France.
Petrach first saw Laura on Good Friday in 1327, when he was 23 years old and she was about 19. That encounter ignited a passion that was to last for more than 20 years and inspired a long series of love poems. Addressed to Laura, Petrarch's collection of 366 poems in her honor was eventually published as the Canzoniere. His writings influenced many authors, including Geoffrey Chaucer (1340?-1400), Edmund Spenser (1552?-1599), and William Shakespeare (1564-1616).
Petrarch's poems express a wide range of emotions, from extreme elation to deep despair. Most of them bear no date and appear collected together according to some baffling, obscure scheme rather than chronologically. Over the centuries, scholars have expended considerable effort trying to bring order to this temporal hodgepodge.
About a decade ago, Frederic J. Jones, then a professor of Italian studies at the University of Wales in Cardiff, conjectured that Petrarch's emotions followed a fairly regular cylical pattern. He performed a detailed linguistic and stylistic analysis of the dated poems, rating the emotions expressed in each on a scale from +1 (ecstatic love) to -1 (deep despair). Intermediate grades corresponded to less extreme feelings, such as simple ardor, serene love, mere friendship, mild melancholy, and anguish.
For example, in sonnet CLXXVI, Petrarch muses:
Parme d'udirla, udendo i rami et l'ore
et le frondi, et gli augei lagnarsi, et l'acque
mormorando fuggir per l'erba verde.
[Her I seem to hear, hearing bough and wind's caress,
as birds and leaves lament, as murmuring flees
the streamlet coursing through the grasses green.]
That fragment rates -0.45, which corresponds to a state of melancholy.
Jones detected six lyrical cycles, spanning about 21 years. He was then able to use that pattern to put the undated poems in some sort of chronological order. Indeed, he went on to claim that his methodology could be applied to any long series of undated love poems.
Stimulated by the Jones analysis, Rinaldi developed a model in which three ordinary differential equations express the emotions of Petrarch and Laura. Laura is described by a single variable L(t), representing her response to the poet at time t. Positive values of L mean affection and encouragement, whereas negative values are associated with coldness and antagonism. Reflecting the greater complexity of Petrarch's personality, two variables describe his state of being: P(t), his love for Laura, and Z(t), poetic inspiration.
The rate of change of Laura's love depends on the sum of terms representing her reaction to Petrarch's expressed love, her response to his physical, social, and intellectual appeal, and some sort of forgetting process. The rate of change of Petrarch's love has analogous termsÑexcept that his response to her appeal also depends on his inspiration function, Z.
"This takes into account the well-established fact that high moral tensions, like those associated with artistic inspiration, attenuate the role of the most basic instincts," Rinaldi notes.
The third equation simply says that Petrarch's love sustains his inspiration, which otherwise would exponentially decay. "In other words," Rinaldi says, "poetic inspiration is an exponentially weighted integral of the passion of the poet for his mistress."
Specifying the values of equation parameters characterizing the personalities and behavior of Petrarch and Laura proved troublesome and highly subjective. "They are only my personal estimates based on the feelings and impressions I had when reading the Canzoniere," Rinaldi admits. Laura, for example, never appears to be strongly involved, while the poet exhibits a tenacious attachment, reinforced by the occasional smile or chance glance from Laura.
Numerically integrating the equations reveals that after an initial, high peak, Petrarch's love tends toward a regular cycle of alternating peaks and valleys. Laura's affection and Petrarch's inspiration show somewhat different patterns.
"At the beginning, Petrarch's inspiration rises much more slowly than his love and then remains positive during the entire period," Rinaldi observes. "This might explain why Petrarch writes his first poem more than three years after he has met her, but then continues to produce lyrics without any significant interruption."
Overall, the results neatly match those obtained independently by Jones.
Rinaldi, however, hasn't been the only one to ponder the link between love affairs and differential equations. About a decade ago, Steven H. Strogatz, now at Cornell University, described how the construction, solution, and interpretation of coupled differential equations representing the evolution of a love affair between two people can be a popular exercise in the college classroom.
Here's a sample scenario: Juliet is in love with Romeo, but Romeo is a fickle lover. The more Juliet loves him, the more he begins to dislike her. But when she loses interest, his feelings for her warm up. She, on the other hand, tends to echo him. Her love grows when he loves her and turns to hate when he hates her.
A simple model of this ill-fated romance generates never-ending cycles of love and hate. All is not lost, however. One-quarter of the time, they manage to bring their love into synchrony for a brief interval of true passion.
"It's a very good way to teach differential equations," Strogatz says. "Students love it."
Love certainly makes the world go round in surprising ways.
Copyright 1998 by Ivars Peterson
References:
Jones, F.J. 1995. The Structure of Petrarch's Canzoniere: A Chronological, Psychological and Stylistic Analysis. Cambridge, England: Boydell & Brewer.
Rinaldi, S. 1998. Laura and Petrarch: An intriguing case of cyclical love dynamics. SIAM Journal of Applied Mathematics 58(August):1205. (Abstract available at http://www.siam.org/journals/siap/58-4/30592.html.)
Strogatz, S.H. 1994. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Reading, Mass.: Addison-Wesley.
______. 1988. Love affairs and differential equations. Mathematics Magazine 61(February):35.
Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.